Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{r^2 - r}{r^2 - 10r + 9}$
First factor the expressions in the numerator and denominator. $ \dfrac{r^2 - r}{r^2 - 10r + 9} = \dfrac{(r)(r - 1)}{(r - 9)(r - 1)} $ Notice that the term $(r - 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r - 1)$ gives: $t = \dfrac{r}{r - 9}$ Since we divided by $(r - 1)$, $r \neq 1$. $t = \dfrac{r}{r - 9}; \space r \neq 1$